Predator-Prey Model: Driven Hunt for Accelerated Grokking

I. A. Lopatin; S. V. Kozyrev; A. N. Pechen

cs.NESep 10, 2025

Predator-Prey Model: Driven Hunt for Accelerated Grokking

Authors: I. A. Lopatin, S. V. Kozyrev, A. N. Pechen

Organizations: Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russia · Ivannikov Institute for System Programming of the Russian Academy of Sciences, Moscow, 109004 Russia

Abstract

A machine learning method is proposed using two agents (representing two coupled replicas of the model parameters) that simulate the biological behavior of a predator and a prey. In this method, the two agents interact with each other via a pair of potentials that simulate the dynamics when the predator chases the prey while the prey runs away from the predator - to perform an optimization on the landscape. This method allows, for optimization landscapes with narrow ravines with gentle slope or even flat bottoms, to avoid optimization getting stuck in a ravine. For some examples of grokking (i.e., delayed generalization) problems we show that this method allows for achieving up to a hundred times faster learning in gradient evaluations and up to 60 times in the wall-clock time compared to the standard learning procedure.

Figures & tables

Figure 1: Typical training processes for ModuloOperation and MNIST with the standard AdamW algorithm. Upper row: for ModuloOperation. Bottom row: for MNIST. In each row from left to right: accuracy of the training model vs epoch, weight norm of the training model vs epoch, effective optimizer step norm

\|\hat{m}_{t}/(\sqrt{\hat{v}_{t}}+\epsilon)\|

for Adam/AdamW vs optimization process step, and distance to the initial model vs optimization process step.

Figure 2: Training after the memorization phase using the PPM in the no Pred. grad mode (no gradient calls for the predator) using the Algorithm 1 . Adam with

\lambda=10^{-2}

is used as the optimizer. Two upper rows are for ModuloOperation and two bottom rows are for MNIST. The 1st column is the prey and predator accuracies for train and test data sets over epochs. The 2nd and the 3rd plots in the upper row for each task (i.e. the second and the third rows) represent Prey model weight norm and norm distance between initial Prey model and current Prey model respectively over steps. The 2nd plots in the 2nd and the 4th rows contain comparison of the optimizer step norm

\|\hat{m}_{t}/(\hat{v}_{t}+\epsilon)\|

for Adam with the prey-predator potential value over steps. We set in ( 3 )

A=150,\sigma=10

for ModuloOperation and

A=15,\sigma=1

for MNIST. For each task we set

\alpha_{p}=2A/3

and

N_{d}=5

Figure 3: A typical example of the PPM standard training process for the ModuloOperation with gradient enabled for the predator. The quantities on the plots are the same as on Fig. 2 .

Figure 4: Training with PP-conn. In both cases we do not call the gradient for the predator. For the MNIST we use exponential moving average (EMA) for the first moment, for the ModuloOperation we do not use it. The quantities on the plots are the same as on Fig. 2 . Comparing with the standard procedure shown in Fig. 1 , PP-conn achieves an acceleration of training up to a hundred times in the number of gradient calls. We perform additional optimizer epochs after achieving grokking phase to show that accuracy does not decrease, as shown in the left column after the vertical dashed green line.

Figure 5: Training for PP-conn with

A>\alpha_{p}

. The quantities on the plots are the same as on Fig. 2 .

Figure 6: Grokking time dependence for addition modulo 139. Left: (in logarithm of the number of epochs) on the sample size

\beta

, which is the proportion of training and test data compared to the maximal possible sample. It shows linear decrease with

\beta

. Right: on the initial weight model norm. It shows linear growth with the weight model norm. Each point is obtained by averaging over 3 runs.

Figure 7: Comparing optimization process for standard Adam algorithm and PPM-conn with optimizer reboots. Upper and bottom rows show Rosenbrock function value (left column) and distance to the minimum (right column). Subplots (a) and (b) for Adam algorithm (with reboots, but for Adam without reboots holds the same trajectory) and subplots (c) and (d) for PPM-conn with reboots.

$\sigma$	$A\backslash\alpha_{p}$	100	500	1000	2000	5000
1	100	1.00, 107.12 (15.48)	x	x	x	x
	200	1.00, 104.72 (13.37)	x	x	x	x
	500	1.00, 106.75 (16.88)	1.00, 90.94 (9.87)	x	x	x
	1000	1.00, 110.05 (16.19)	1.00, 71.43 (7.92)	1.00, 102.15 (11.30)	x	x
	2000	1.00, 113.79 (15.85)	1.00, 67.00 (8.64)	1.00, 62.09 (10.03)	0.98, 185.37 (43.19)	x
	5000	1.00, 119.31 (17.54)	1.00, 75.13 (10.61)	1.00, 50.17 (10.14)	–	0.99, 150.29 (36.52)

Table 1: Ablation for Modulo Operation for various values of the parameters of PPM-conn.

$\sigma$	$A\backslash\alpha_{p}$	10	50	100	500	1000
1	10	1.00, 80.30 (10.35)	x	x	x	x
	50	1.00, 88.48 (13.20)	1.00, 77.17 (10.03)	x	x	x
	100	1.00, 92.28 (14.82)	1.00, 101.63 (15.86)	1.00, 112.99 (20.51)	x	x
	200	1.00, 99.76 (19.51)	1.00, 128.66 (27.36)	0.98, 175.34 (37.48)	x	x
	500	0.95, 131.08 (36.49)	0.90, 191.33 (41.03)	–	1.00, 114.88 (17.96)	x
	1000	0.77, 185.72 (46.96)	–	–	–	1.00, 119.08 (18.92)

Table 2: Ablation for MNIST for various values of the parameters of PPM-conn.

Figure 8: Histograms for ablation. Upper row shows acceleration in epochs for PPM-conn compared with standard AdamW for (a) ModuloOperation and (b) MNIST. Bottom row shows the number of epochs for achieving grokking with PPM-conn for (c) ModuloOperation and (d) MNIST. Parameters for PPM-conn have been chosen as best parameters according to ablation procedure, i.e. tables 1 and 2 for ModuloOperation and MNIST tasks respectively.

Figure 9: Curriculum learning for modular division task. Sudden drops in train accuracy are caused by changing

p_{j}

Optimizer	Modulo Operation	MNIST
NAdam, default params.	$2\cdot 10^{3}$	$1\cdot 10^{4}$
NAdam, lr $=10^{-3}$ , wd $=10^{-2}$	—	$8\cdot 10^{2}$
RAdam, default params.	$5\cdot 10^{3}$	—
RAdam, lr $=10^{-3}$ , wd $=10^{-2}$	—	$9\cdot 10^{2}$
Lion, lr $=10^{-3}$ , wd $=10^{-2}$	$4\cdot 10^{3}$	—
RMSprop, default params.	—	—

Table 3: Comparing different optimizers

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Department of Machine Learning and Neural Computing Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen, the Netherlands

Dynamical Systems Agent-Based Modeling

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Recent growth in compute power has made it increasingly feasible to use large-scale agent-based models to simulate complex adaptive systems. A central difficulty is that such models contain many local rules and parameters, where small changes can lead to runaway behaviour, population collapse, or saturation at artificial bounds. We study this problem in a continuous predator-prey system where sheep and wolves are active agents with local sensing, internal energy, and recurrent neural network-based controllers. We ask whether environmental and demographic parameters can be tuned so that the resulting population dynamics resemble classical Lotka-Volterra cycles. We optimise these parameters with a feature-based loss that rewards sustained oscillations, phase lag, bounded populations, and long-term persistence, first for random controllers and then for evolved controllers in a more naturalistic setting. The model is implemented in ABMax, a JAX-based agent-based modelling framework that enables efficient batched simulation on hardware accelerators.

Abstract

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